Elektromanyetik integral denklemlerin hızlı ve doğru çözümleri için yeni yöntemler

As the scale and complexity of electromagnetic scattering problems continue to grow, solving frequency-domain integral equations (IEs) with high accuracy and computational efficiency presents increasing challenges. To address these, two novel methods are proposed, each offering robust alternatives to conventional approaches such as the multilevel fast multipole algorithm (MLFMA), which often suffers from the low-frequency breakdown (LFB) problem and inefficiencies in complex or elongated geometries. The first method introduces a broadband and highly parallelizable framework based on machine learning for accelerating the iterative solution of frequency-domain IEs. Far-zone interactions are evaluated through a group-by-group interaction model using a one-box buffer scheme. The scatterer is divided into uniformly sized boxes, and each subdomain basis functions within boxes are represented by a fixed number of Hertzian dipoles regarded as uniform basis functions. Neural networks trained on the dipole Green's function are employed to model far-field interactions, enabling accurate and reusable computations independent of problem geometry, provided that box sizes and relative distances are fixed. This framework eliminates the LFB problem and provides strong scalability in parallel environments. Its accuracy and efficiency are validated through comparisons with MLFMA and Mie series results, along with parallel scalability benchmarks. The second method focuses on improving far-zone interaction efficiency through a hierarchical strategy that incorporates Taylor expansions and virtual box decomposition. While maintaining the hierarchical tree structure of MLFMA, each box of size a is subdivided into virtual boxes of size sqrt(a), and interactions are approximated using Taylor expansions of the reduced Green's function. This setting ensures that the number of expansion coefficients within virtual boxes remains constant with increasing box size, preserving O(N log N) complexity even for electrically large or quasi-one-dimensional geometries where MLFMA struggles. Unlike matrix-decomposition-based methods, the proposed scheme relies entirely on algebraic operations. The method's robustness is demonstrated through numerical simulations that confirm its accuracy and efficiency relative to MLFMA.